Bed sector division

A rectangular bed 3960 cm long and 825 cm wide needs to be divided into several equal square sectors on which scientists will test different fertilizers. Into what is the smallest number of sectors this bed can be divided?

Final Answer:

n =  120

Step-by-step explanation:

$$\begin{gathered} a=3960\text{cm} \\ b=825\text{cm} \\ 3960=2^3\cdot 3^2\cdot 5\cdot 11 \\ 825=3\cdot 5^2\cdot 11 \\ \text{GCD}(3960,825)=3\cdot 5\cdot 11=165 \\ x=GCD(a,b)=GCD(3960,825)=165\text{cm} \\ {S}_1=a\cdot b \\ {S}_2=x\cdot x \\ n=\frac{S_1}{S_2}=\frac{a\cdot b}{x\cdot x}=\frac{a}{x}\cdot \frac{b}{x}=n_1\cdot n_2 \\ {n}_1=a/x=3960/165=24 \\ {n}_2=b/x=825/165=5 \\ n=n_1\cdot n_2=24\cdot 5=120 \end{gathered}$$

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